[1] My 3xBB bet range here would be {99+,ATs+,KTs+,QTs+,JTs,75s-74s,64s+,53s+,43s,AJo+,KQo}

Some of these hands typically would be limped if I didn't know the BU was a nit, but I'd raise to try and keep him out of the pot/isolate him or the blinds for HU. I'd also limp other PP's and some other random connectors since most often the BU is going to fold anyway.

Since the BU called I can assume that his range of hands has become significantly smaller, most likely JJ+, AQo+, AJs+, maybe small suited connectors, but this is highly situational depending on how he acts on the flop and I'll re-evaluate then.

[2] Since I was the original aggressor I would put in a c-bet with the majority of my range since most of the villain's assumed range isn't hit. So here I would continue with
{99+,AQs+,ATs,KQs,KTs,QTs,JTs,74s,64s+,53s+,AKo}

With the villains call on the flop I would tend to become a bit more worried about what hands he could have and would put him on either a straight draw, an overpair, or I would rethink about his range and possibly give him a set of tens. Given his nature of fit-or-fold most likely a set, but I'd wait to see what the turn brings.

[3] At this point since villain has called, the majority of my range turns into a check and hoping for the free card. My remaining betting hands are {KK+,JJ-TT,JTs} The remainder of cards I have to bet with here will cover most of villain's range as far as overpairs are concerned, JJ and TT give me a nice set on the turn (TT beforehand), and JTs give a nice two pair, also lessening the chances that villain has made a set with either of those cards since there are only two combinations of cards that would give him sets with either tens or jacks and would defeat his overpair.

[4] In this spot I would check with AA and KK because they have lost a lot of value with the pair of jacks on the board so those have been eliminated out of my range. Since {JJ-TT, JTs} would all give me boats I would go ahead and bet out on all of them and just hope for the best. The bet size makes sense in this case since we most likely have the nuts and we'd like it called rather than to scare him off with too large a size bet.